Section 13

Complex numbers

fundamentalviz

i^{2} = - 1; z = a + bi

Cartesian representation of a complex number.

fundamentalviz

\overset{z}{ˉ} = a - bi; z \overset{z}{ˉ} = ∣ z ∣^{2}

Reflects a complex number across the real axis.

fundamentalviz

∣ z ∣ = a^{2} + b^{2}

Distance from the complex number to the origin.

intermediateviz

z = r (cos θ + i sin θ) = r e^{i θ}

Represents a complex number by magnitude and angle.

intermediateviz

e^{i θ} = cos θ + i sin θ

Connects complex exponentials with trigonometry.

intermediateviz

(cos θ + i sin θ)^{n} = cos (n θ) + i sin (n θ)

Raises complex numbers in polar form to a power.

intermediateviz

w_{k} = r^{1/ n} e^{i (θ + 2 π k) / n}

The roots are distributed uniformly on a circle.